In [1]}, p. 11, the embedding
\(i_A^X \colon A \rightarrow X\)  is omitted for simplicity, and
a morphism \(u\)  is just a function \(u^0 \colon X \rightarrow Y\)  such that
\(\forall _{a \in A}\exists _{b \in B}\big (u^0(i_A^X(a)) =_Y i_B^Y(b)\big ).\) 
